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December 2015 version

Posted by ravivakil under

2015-16 course,

Actual notes
[27] Comments
My goal of posting current versions every month almost worked. The December 29, 2015 version is in **the usual place**. (But I am only posting it on January 1, 2016 — happy new year!)

As usual, there are many small changes, but nothing that should particularly make you want to download it if you have the previous version. And as always, I have a big list of emails, responses, etc. that I want to respond to, and a number I intend to respond to fairly soon.

January 13, 2016 at 11:47 am

By the way, whatever happened to the warnings at the start of the April 2015 version? Did all of that get resolved?

February 4, 2016 at 11:38 am

Some but not all — but anything not resolved is still on the to-do list!

January 22, 2016 at 4:23 pm

Two more tiny suggestions:

* In the proof of 19.4.1, I don’t think it makes sense to say that a map is “base-point-free”.

* Right after the proof of Claim 19.7.1 I think the displayed equation should just say 1 and not “1 or 0”. The latter can’t happen and is never mentioned again. Right after this I think you need one more invocation of Riemann-Roch (and not just Serre duality) in order to justify .

January 25, 2016 at 8:53 am

[Sorry for all the little posts.] I think it would be good to have an exercise, maybe near the start of 11.3 where it’s first needed, on relating the dimension of a projective $k$-scheme to its cone. Something along of the lines of Hartshorne’s exercises I.2.6 and I.2.10. This seems crucial for 11.3.C, for example.

February 4, 2016 at 11:41 am

Agreed! I’ve now added it to the to-do list. The dimension theory exposition around the start of 11.3 is going to change soon, so I’ll hold off making this particular change.

January 26, 2016 at 8:57 pm

It seems unfortunate to use the notation to mean for most of the book, particularly when talking about ramification, and then switch to using it for the map . Is this intentional?

January 26, 2016 at 9:04 pm

I don’t think it was — can you remind me where I said this? (And thanks for all the little posts — they are not disappearing into a black hole!)

January 27, 2016 at 6:16 am

Sorry, I should have been more specific. The second usage of comes up when you discuss generic smoothness, right after 25.3.4. I’ve seen people (Mumford?) use for this although maybe that’s bad too.

No worries, by the way.

February 4, 2016 at 11:35 am

That is a good point! I’m just going to remove that paragraph. I had wanted to say something conceptual, but saying it well would just end up confusing everyone.

February 10, 2016 at 2:47 pm

In the proof of Theorem 22.3.2 the notation for the Rees algebra comes out of nowhere, I think. This is what had been parenthetically called earlier but the reader has probably forgotten that by this point.

Have people found this proof confusing, by the way? I can’t tell whether it’s my ignorance or whether something is really off with it. I know that’s vague, sorry.

February 10, 2016 at 2:48 pm

Also, it might make sense to change either the discussion right before this proof (which uses for the ring) or the notation of the proof. That’s more of a nitpick.

March 3, 2016 at 3:11 pm

In 9.5.7, I think you omitted the first part of your parenthetical definitionâ€”it should read

…is purely inseparable (i.e., any has minimal polynomial over …

rather than

…is purely inseparable polynomial over …

Interestingly, this omission had been included in earlier versions of the book, but it’s missing now.

March 3, 2016 at 6:31 pm

In the proof of 9.5.23.(b), I think it should read

… Then use Exercise 9.5.K.

instead of

… Then use Proposition 9.5.K.

March 4, 2016 at 7:33 pm

Thanks, now fixed! (There are a bunch of things in 9.5 that need fixing, some of which are on my to-do list.)

March 9, 2016 at 8:29 pm

I don’t know if you were thinking of this, but (at least for me) the tensor-finiteness trick isn’t as immediately understandable as everything else in the book (which is really just a positive statement about how understandable the book is!). Though I personally rationalized away this difficulty using the fact that the latter half of 9.5 is double-starred anyway.

March 5, 2016 at 4:17 pm

I think it would be good to clarify what the multiplication in Exercise 16.7.I is supposed to be.

March 29, 2016 at 10:26 am

Good point! I did this in the form of a query, by no longer claiming the exericse is easy. I have added: The tricky part of this exercise is figuring out how to interpret multiplication (of Plucker coordinates) here.

March 6, 2016 at 4:23 am

A typo in 2.6.2 (construction of the inverse image): $\pi^{-1} G_{pre}$ should be $\pi^{-1}_{pre} G$.

Also, I found the phrase “fully faithful subcategory” in Exercise 1.5.H, which could simply be replaced by “full subcategory”. Same in Exercise 9.1.C and in the Index.

April 24, 2016 at 11:59 am

Very minor comment:

The first sentence on page 59 (Section 1.7.1 Double complexes) reads —

“There are variations on this definition, where for example the vertical arrows

go downwards, or some different subset of the E^{p,q} are required to be zero, ….”

However, in the discussion until that point, you did not require any of the E^{p,q} to be zero. (The zero-outside-first-quadrant assumption only appears in the next page.)

July 29, 2016 at 11:24 pm

The first of a string of comments, spilling over from my e-mail. I’ll put them separately to make addressing them separately easier.

In theorem 17.4.3, to get an irreducible regular projective curve from an integral curve of finite type, you normalise and then take a regular projective compactification. I wasn’t sure why it should be the case that the normalisation was separated, (which was one of the conditions for finding a projective normalisation – theorem 17.4.2) since we didn’t assume our original curve to be separated. Should separatedness be a condition for the curves of type iv), or is the normalisation automatically separated for some reason?

July 29, 2016 at 11:29 pm

I think that exercise 17.4.C should have the curves be integral rather than just irreducible, so that we can talk about the function fields of the curves and the degree of the map.

July 29, 2016 at 11:37 pm

Exercise 17.4.F asks about a degree d line bundle on a curve – but I don’t think that this has been defined so far, so I think maybe it should be put somewhere else (maybe in 18.4) unless you mean to use the characterisation of degree from the previous remark? Or something else?

July 29, 2016 at 11:40 pm

On page 472, I think the equation label (18.3.2.2) has been parsed wrong, it looks a lot smaller than usual.

July 30, 2016 at 12:36 pm

Just a comment: Exercise 15.4.D part b) (showing that the saturation map is an isomorphism in large degree) seems like an unusually hard exercise, particularly as only a part of a question rather than as a standalone, and without at (hard exercise) label. I’m quite possibly missing something but I couldn’t work out a short proof, and the only one I could find adapts a proof from Hartshorne, see e.g. here:

http://math.stackexchange.com/questions/1147772/hartshorne-ii-ex-5-9a-or-r-vakil-ex-15-4-db-saturated-modules

The proof presented here is pretty involved and I certainly don’t think I would have come up with it by myself. Is there an easier way to do it that you had in mind? I seem to recall that it’s not hard to show that the associated sheaves are isomorphic, or that the sheaf associated to the cokernel of the saturation map is 0. It follows that if the saturated module is finitely generated, then the cokernel is 0 in high degree which would prove the claim, but this seems hard to prove.

July 30, 2016 at 12:58 pm

Hopefully my final comment for a while!

Exercise 17.3.B which shows that the composition of projective morphisms is projective, where the final target is quasi-compact has me a little confused. It seems from the hints that you’re aiming to find a line bundle that is locally O(1) and then use this to show that it is globally O(1). The affine case shows that L x (pi)_* M^m is locally very ample for all large m. Then by quasicompactness we can find an m that works for each patch in a finite affine open cover, so the trick then is to show that L x (pi)_* M^m is globally O(1).

You mention several times throughout 17.3 that in section 17.3.4 we will see that in Noetherian circumstances, projectivity with a choice of O(1) is affine local. It seems that in this exercise we’re trying to prove that it is affine local, but without the Noetherian hypotheses.

I asked about it here:

http://math.stackexchange.com/questions/1863379/how-to-complete-vakils-proof-that-the-composition-of-projective-morphisms-are-p

But so far the only answer seems to be going along the lines of the argument for affine-localness of relative very-ampleness for locally Noetherian schemes in the double starred section of relative very ampleness later in the chapter, which may or may not work out without Noetherian hypotheses. I also saw your original mathoverflow post about this question, where a few people remarked that EGA and the stacks project only seem to prove this proposition for the case where Z is additionally quasi-separated, although no-one disagreed with your claim that quasi-compactness is enough.

I guess the question is, then: is it really true that we don’t need further quasi-separated/locally Noetherian hypotheses? If so, would you mind clarifying how the local to global argument works?

(I’m sorry to bother you with questions asking for clarifications about exercises, but I’ve given them a lot of thought and spent a good deal of time looking for answers externally without much success. It’s therefore hard for me to tell if the intended hints don’t work, or if I’ve just not worked out how to apply them properly.)

September 13, 2016 at 9:27 am

I think that you got the definition of the Frobenius wrong in Exercise 7.3.R. It shouldn’t just take each x_i to its pth power, but also elements of k to their pth powers.

October 11, 2016 at 8:59 am

In the remark after Exercise 29.2.B, you cite the Krull intersection theorem for showing that local rings are separated with respect to their maximal ideals, though this also requires the local ring to be Noetherian. So it should read “If (A,m) is a Noetherian local ring…”